These words are rather close synonyms. Below are imperfective and perfective forms of both (perfective form implies that the action is completed):

Изучать (imp)/изучить (perf)
Учить (imp)/выучить (perf)

"Изучать" means to study something seriously or to research.

Я изучаю математику - I study mathematics (for example, this person is a Math major)

"Учить" is often used as an equivalent of "изучать" in informal speech, but generally it means that you learn something at the moment or work on some specific task.

Я учу математику - I('m) learn(ing) math (a homework, for example)
Я выучил математику ~ I've done my Math homework (lterally "I've learnt math")

In other words "учить" and "изучать" can be interchangable in certain situations (the nuances are still there, but they can be neglected):
Я учу французский
Я изучаю французский

While sometimes it's only one or the other:

Я изучил эту проблему - I've studied/researched this issue
You can't say "Я выучил эту проблему", it will sound as if a problem is something you can learn by heart.

Я выучил это стихотворение - I've learnt this poem (by heart)
Я изучил это стихотворение - I've researched this poem, I've learnt a lot about it (its structure, etc.)